PIR Codes with Short Block Length

In this work private information retrieval (PIR) codes are studied. In a $k$-PIR code, $s$ information bits are encoded in such a way that every information bit has $k$ mutually disjoint recovery sets. The main problem under this paradigm is to minimize the number of encoded bits given the values of $s$ and $k$, where this value is denoted by $P(s,k)$. The main focus of this work is to analyze $P(s,k)$ for a large range of parameters of $s$ and $k$. In particular, we improve upon several of the existing results on this value.Comment: 10 pages, 1 tabl

Limit sets of stable Cellular Automata

We study limit sets of stable cellular automata standing from a symbolic dynamics point of view where they are a special case of sofic shifts admitting a steady epimorphism. We prove that there exists a right-closing almost-everywhere steady factor map from one irreducible sofic shift onto another one if and only if there exists such a map from the domain onto the minimal right-resolving cover of the image. We define right-continuing almost-everywhere steady maps and prove that there exists such a steady map between two sofic shifts if and only if there exists a factor map from the domain onto the minimal right-resolving cover of the image. In terms of cellular automata, this translates into: A sofic shift can be the limit set of a stable cellular automaton with a right-closing almost-everywhere dynamics onto its limit set if and only if it is the factor of a fullshift and there exists a right- closing almost-everywhere factor map from the sofic shift onto its minimal right- resolving cover. A sofic shift can be the limit set of a stable cellular automaton reaching its limit set with a right-continuing almost-everywhere factor map if and only if it is the factor of a fullshift and there exists a factor map from the sofic shift onto its minimal right-resolving cover. Finally, as a consequence of the previous results, we provide a characterization of the Almost of Finite Type shifts (AFT) in terms of a property of steady maps that have them as range.Comment: 18 pages, 3 figure

Constructions of Batch Codes via Finite Geometry

A primitive $k$-batch code encodes a string $x$ of length $n$ into string $y$ of length $N$, such that each multiset of $k$ symbols from $x$ has $k$ mutually disjoint recovering sets from $y$. We develop new explicit and random coding constructions of linear primitive batch codes based on finite geometry. In some parameter regimes, our proposed codes have lower redundancy than previously known batch codes.Comment: 7 pages, 1 figure, 1 tabl

Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes

We propose three private information retrieval (PIR) protocols for distributed storage systems (DSSs) where data is stored using an arbitrary linear code. The first two protocols, named Protocol 1 and Protocol 2, achieve privacy for the scenario with noncolluding nodes. Protocol 1 requires a file size that is exponential in the number of files in the system, while Protocol 2 requires a file size that is independent of the number of files and is hence simpler. We prove that, for certain linear codes, Protocol 1 achieves the maximum distance separable (MDS) PIR capacity, i.e., the maximum PIR rate (the ratio of the amount of retrieved stored data per unit of downloaded data) for a DSS that uses an MDS code to store any given (finite and infinite) number of files, and Protocol 2 achieves the asymptotic MDS-PIR capacity (with infinitely large number of files in the DSS). In particular, we provide a necessary and a sufficient condition for a code to achieve the MDS-PIR capacity with Protocols 1 and 2 and prove that cyclic codes, Reed-Muller (RM) codes, and a class of distance-optimal local reconstruction codes achieve both the finite MDS-PIR capacity (i.e., with any given number of files) and the asymptotic MDS-PIR capacity with Protocols 1 and 2, respectively. Furthermore, we present a third protocol, Protocol 3, for the scenario with multiple colluding nodes, which can be seen as an improvement of a protocol recently introduced by Freij-Hollanti et al.. Similar to the noncolluding case, we provide a necessary and a sufficient condition to achieve the maximum possible PIR rate of Protocol 3. Moreover, we provide a particular class of codes that is suitable for this protocol and show that RM codes achieve the maximum possible PIR rate for the protocol. For all three protocols, we present an algorithm to optimize their PIR rates.Comment: This work is the extension of the work done in arXiv:1612.07084v2. The current version introduces further refinement to the manuscript. Current version will appear in the IEEE Transactions on Information Theor